could i have help on parts B and C of this assignment please. Its really urgent :s
f(t) = 5 e^(-0.8 t) sin(2 pi t / 3)
(b) find f'(t).
Use the product rule:
f'(t) = 5 d/dt[e^(-0.8 t)] sin(2 pi t /3) + 5 e^(-0.8 t) d/dt[sin(2 pi t / 3)]
......= 5 (-0.8) e^(-0.8) sin(2 pi t /3) + 5 e^(-0.8 t) (2 pi / 3) cos(2 pi t / 3)
......= 5 e^(-0.8 t) {(-0.8) sin(2 pi t /3) + (2 pi / 3) cos(2 pi t / 3)}
(c) Solve f'(x) = 0 to determine the maximum deflection and when it occurs.
f'(t) = 0 after cancelling the leading intrinscily non zero term gives:
(-0.8) sin(2 pi t /3) + (2 pi / 3) cos(2 pi t / 3) = 0
or:
tan( 2 pi t / 3) = 2 pi / 2.4,
so:
2 pi t / 3 =arctan(2 pi / 2.4) ~= 1.206 + pi n, n=0, 1, 2, ...
(only positive n will be considered as usually t is measured from the time
of the disturbance to the system, so times <0 are meaningless) so:
t = 0.573 + 3/2 n, n=0, 1, 2, ..
Now as e^(-0.8 t) is a strictly decreasing function the deflection of
maximum amplitude is the first of these, so it occurs at t=0.573, and is of
amplitude:
5 e^(-0.8*0.573) sin(2 pi*0.573 / 3) = 2.95
RonL